We experimentally investigate the morphology and breakup of a droplet as it descends freely from a height and encounters an airstream. The size distributions of the child droplets are analyzed using high-speed shadowgraphy and in-line holography techniques. We find that a droplet falling from various heights exhibits shape oscillations due to the intricate interplay between inertia and surface tension forces, leading to significant variations in the radial deformation of the droplet, influencing the breakup dynamics under an identical airstream condition. Specifically, the droplet undergoes vibrational breakup when introduced at a location slightly above the air nozzle. In contrast, as the release height of the droplet increases, keeping the Weber number defined based on the velocity of the airstream fixed, a dynamic interplay between the inertia of the droplet and the aerodynamic flow field comes into play, resulting in a sequence of breakup modes transitioning from vibrational breakup to retracting bag breakup, bag breakup, bag-stamen breakup, retracting bag-stamen breakup, and eventually returning to vibrational breakup. Our experiments also reveal that the size distribution resulting from retracting bag breakup primarily arises from rim and node fragmentation, leading to a bimodal distribution. In contrast, bag and bag-stamen breakups yield a trimodal size distribution due to the combined contributions of bag, rim, and node breakup mechanisms. Furthermore, we utilize a theoretical model that incorporates the effective Weber number, considering different release heights. This model accurately predicts the size distribution of the child droplets resulting from the various breakup modes observed in our experiments.

The effect of viscosity contrast between a jet and its surroundings is experimentally investigated using density-matched fluids. A gravity-driven flow is established with a jet of saltwater emerging into an ambient medium composed of high-viscosity propylene glycol. Jet Reynolds numbers, $\mathrm{Re}$, ranging from 1600 to 3400 were studied for an ambient-to-jet viscosity ratio, $M$, between 1 and 50. Visualization suggests that at low values of the viscosity ratio, the jet breakdown mode is axisymmetric, while helical modes develop at high values of viscosity ratio. The transition between these two modes is attempted to be delineated using a variety of diagnostic tools. Hot-film anemometry measurements indicate that the onset of the helical mode is accompanied by the appearance of a discrete peak in the frequency spectrum of velocity fluctuations, which exhibits little spatial variation for the first several diameters in the downstream direction. Laser-induced fluorescence (LIF) is used to identify the jet boundary against the background. An analysis of high-speed images acquired using the LIF technique enables identification of the spatial growth rate of waves on the jet boundary, as well as the frequency of oscillation of the weakly diffusive interface. Temporal fluctuations of fluorescence intensity are found to be spatially invariant in the jet near field, further attesting to behavior consistent with that of a self-sustained oscillation whose frequency depends on the viscosity ratio. The observed frequencies show trends similar to those of absolutely unstable modes calculated from spatiotemporal linear stability theory presented in a companion paper. Spectral proper orthogonal decomposition was used to analyze the images and identify the various spatial modes, and suggests the existence of a single dominant mode. Together, these observations provide strong circumstantial evidence for the existence of a global mode that arises from the absolute instability of velocity and viscosity profiles in a region close to the nozzle exit plane.

We investigate the perturbation dynamics in a supersonic shear layer using a combination of large-eddy simulations (LES) and linear-operator-based input-output analysis. The flow consists of two streams—a main stream (Mach 1.23) and a bypass stream (Mach 1.0)—separated by a splitter plate of nonnegligible thickness. We employ spectral proper orthogonal decomposition to identify the most energetic coherent structures and bispectral mode decomposition to explore the nonlinear energy cascade within the turbulent shear-layer flow. Structures at the dominant frequency are also obtained from a resolvent analysis of the mean flow. We observe higher gain at the dominant frequency in resolvent analysis, indicating the dominance of Kelvin-Helmholtz (KH) instability as the primary disturbance energy-amplification mechanism. To focus on realizable actuator placement locations, we further conduct an input-output analysis by restricting a state variable and spatial location of an input and output. Various combinations of inputs and output indicate that the splitter plate trailing surface is the most sensitive location for introducing a perturbation. Upper and lower surface inputs are less influential in modulating wavepackets in the shear layer but introduce pressure instability waves in the main and bypass streams, respectively. The analysis reveals that the phase speed of pressure waves depends on the state variable and input location combination. For all combinations, the KH instability plays a key role in amplification, which reduces significantly as the input location is moved upstream relative to the splitter plate trailing edge. Furthermore, two-dimensional nonlinear simulations with unsteady input at the upper surface of the splitter plate show remarkable similarities between pressure modes obtained through dynamic mode decomposition and those predicted from linear input-output analysis at a given frequency. This study emphasizes the strength of linear analysis and demonstrates that predicted coherent structures remain active in highly nonlinear turbulent flow. The insights gained from the input-output analysis can be further leveraged to formulate practical flow control strategies.

Fast and accurate predictions of turbulent flows are of great importance in the science and engineering field. In this paper, we investigate the implicit U-Net enhanced Fourier neural operator (IUFNO) in the stable prediction of long-time dynamics of three-dimensional (3D) turbulent channel flows. The trained IUFNO models are tested in the large-eddy simulations (LES) at coarse grids for three friction Reynolds numbers: ${\text{Re}}_{\tau }\approx 180$, 395, and 590. The adopted near-wall mesh grids are tangibly coarser than the general requirements for wall-resolved LES. Compared to the original Fourier neural operator (FNO), the implicit FNO (IFNO), and U-Net enhanced FNO (UFNO), the IUFNO model has a much better long-term predictive ability. The numerical experiments show that the IUFNO framework outperforms the traditional dynamic Smagorinsky model and the wall-adapted local eddy-viscosity model in the predictions of a variety of flow statistics and structures, including the mean and fluctuating velocities, the probability density functions (PDFs) and joint PDF of velocity fluctuations, the Reynolds stress profile, the kinetic energy spectrum, and the Q-criterion (vortex structures). Meanwhile, the trained IUFNO models are computationally much faster than the traditional LES models. Thus, the IUFNO model is a promising approach for the fast prediction of wall-bounded turbulent flow.

In Navier-Stokes turbulence, a bottleneck effect in the energy cascade near the viscous cutoff causes an overshoot in the energy spectrum, or spectral bump, relative to Komogorov's −5/3 power-law scaling. A similar spectral overshoot occurs in large-eddy simulations (LES) when an eddy viscosity model is used. It is not a viscous phenomenon, but rather is caused by error in the residual stress model. This artificial bottleneck effect in LES leads to an over-prediction of kinetic energy even if a reliable dynamic procedure is used to accurately capture the spectral decay at the cutoff length scale. Recently, Johnson [J. Fluid Mech. 934, A30 (2022)] introduced a physics-inspired generalization of the concept of spatial filtering that provides a dynamic procedure that does not require a test filter calculation. In this paper, this method of Stokes flow regularization (SFR) is used alongside fundamental considerations related to kinetic energy to generate a range of LES models to explore the artificial bottleneck effect in more detail. The coefficients for each dynamic model are determined locally, without the need of averaging over homogeneous directions. The theory directly provides stabilizing elements such as local averaging of coefficients. A posteriori tests of the models in isotropic turbulence are reported, demonstrating the robustness of the SFR-based dynamic procedure for a range of model forms and providing a framework for fair comparisons between them in terms of their impact on the bottleneck effect. An effective means of mitigating the bottleneck effect is to introduce a nonlinear gradient component in the residual stress closure, forming a dynamic mixed model. One primary reason for the efficacy of this approach is that the nonlinear gradient model is able to accurately capture aspects of the local structure of the residual stresses, leading to a better representation of energy cascade efficiencies.

Gaseous detonation waves in a uniform mixture have been studied widely, but uniformity is seldom realized in practical applications such as detonation-based engines. Nonideal scenarios involving incomplete mixing and curved intake compression lead to the thermal stratification of reactants. Local high-temperature regions first trigger the reactant autoignition and even result in the untimely formation of a detonation wave. Using the two-dimensional Euler equations and a detailed ${\mathrm{H}}_2$–air reaction mechanism, we examine the effects of reactant thermal stratification on the autoignition wave morphology in hypersonic reactive flows. Three flow regimes, namely, the autoignition-driven reaction front, detonation wave, and decoupling shock/reaction front, are observed. These flow regimes are determined by the temperature gradient, and only a moderate temperature gradient can trigger an oblique detonation wave. The oblique detonation can stabilize in hypersonic inflows, primarily because the upstream autoignition region acts as an anchorage point. Comparisons of one- and two-dimensional autoignitions confirm that supersonic flow enhances the formation of pressure waves. An analysis of the reaction front propagation speeds reveals that the detonation development is determined by two aspects. One is the convergence of compression waves originating from thermal expansion and supersonic flows, and another is the reaction front propagation speed at the early stage, which must exceed the local sound speed to promote positive feedback between pressure waves and heat release.

The varicose dynamics of a forced gravitational liquid sheet (curtain) issuing into a quiescent gaseous ambient is numerically investigated in this work. The study is relevant for technological applications such as coating deposition, where varicose perturbations of the curtain interfaces can arise due to axial velocity fluctuations coming from the delivering pump placed upstream of the coating die. The investigation is performed in supercritical regime, namely, for Weber number $\mathrm{We}>1$. Two methodologies are employed: a simplified one-dimensional (1D) linear model and two-dimensional (2D) volume-of-fluid simulations. Using harmonic forcing perturbations of the streamwise velocity applied at the inlet section, the curtain varicose dynamics is excited by varying the forcing frequency $f$ and amplitude $A_u$ of the perturbations for different values of $\mathrm{We}$. As a significant result, the 1D analysis reveals that the curtain oscillations amplitude reaches a maximum value for a certain forcing frequency $f=f_{\max }$. In other terms, it is found that the flow manifests a resonance behavior, with the oscillation frequency $f_{\max }$ and corresponding amplitude $A_{h,\max }$ both scaling as ${\mathrm{We}}^{1/3}$, while the average wavelength ${\overline{\lambda }}_{\max }$ scales as ${\mathrm{We}}^{-1/3}$. These scaling laws are confirmed both by theoretical insights and 2D simulations. Moreover, it is found that the 2D curtain breaks up numerically by increasing the forcing amplitude $A_u$. The numerical rupture is determined by a progressive curtain thinning induced by the varicose deformation, which moves upstream by increasing $\mathrm{We}$, i.e., downstream by increasing the surface tension coefficient. In this respect, surface tension is found to play a stabilizing role on the varicose oscillations of the curtain.

Acoustic streaming shows great potential in applications such as bubble dynamics, cell aggregation, and nanosized particle isolation in the biomedical and drug industries. As the acoustic shock distance decreases with the increase of incident frequency, the nonlinear propagation effect will play a role in acoustic streaming, e.g., Eckart (bulk) streaming at a few gigahertz. However, the theory of source terms of bulk streaming is still missing at this stage when high-order acoustic harmonics play a role. In this paper, we derive the source term including the contribution of high-order harmonics. The streaming-induced hydrodynamic flow is assumed to be incompressible and no shock wave occurs during the nonlinear acoustic propagation as restricted by the traditional Goldberg number $\mathrm{\Gamma }<1$ or $\mathrm{\Gamma }\approx 1$, which indicates the importance of nonlinearity relative to dissipation. The derived force terms allow evaluating bulk streaming with high-order harmonics at gigahertz and provide an exact expression compared to the existing empirical formulas. Numerical results show that the contribution of higher-order harmonics increases the streaming flow velocity by more than $20\%$. Our approach clearly demonstrates the errors inherent in the expression introduced by Nyborg which should be avoided in numerical computations as it includes part of the acoustic radiation force that does not lead to acoustic streaming.

When a drop impacts next to the edge of a solid substrate, it may spread beyond this edge. It then forms a liquid sheet surrounded by a rim from which droplets may be ejected. This work investigates the influence of the edge shape on the rim dynamics and subsequent droplet ejections. Experiments of drop impacts on star-shaped poles are reported. Both the rim and the ejected droplets are tracked. An analytical model is proposed to rationalize the amplitude of rim deformations induced by the edge shape. Statistical distributions of position, size, and velocity of ejected droplets are also shaped by the edge geometry.

A numerical experiment is conducted to investigate the response of a homogeneous isotropic turbulent field at a statistically equilibrium state when the energy cascade process is abruptly interrupted. Vortex motions of a certain scale in the inertial subrange are extracted using a Fourier bandpass filter and forcibly damped by applying artificial forces to the small regions that are the target vortices. Once the forces are applied, the target vortices immediately disappear from the flow field, which is followed by a slight increase in kinetic energy in the larger scale range and a decrease in the smaller scale range. The decrease in energy in the smaller scale range is likely to be caused by the decrease in the stretching speeds of the vortices of that range. Next, the behaviors of individual vortices whose scales are either four times or twice as large as the target scale are tracked using a method in which each vortex is reconstructed as a group of vortex units. It is found that the vortices that are twice as large as the target vortices show smaller curvatures and longer lifespans in comparison to the case without artificial forces, while no remarkable changes are found for the vortices that are four times larger.

The mucus on the bronchial wall forms a thin layer of non-Newtonian fluid, protecting the lungs by capturing inhaled pollutants. Due to the corrugation of its interface with air, this layer is subject to surface tension forces that affect its rheology. This physical system is analyzed using lubrication theory and three-dimensional simulations. We characterize the nonlinear behavior of the mucus and show that surface tension effects can displace overly thick mucus layers in airway bifurcations. This movement can disrupt the mucociliary clearance and break the homogeneity of the layer thickness.

Pendant drops spontaneously appear on the underside of wet surfaces through the Rayleigh-Taylor instability. These droplets are connected to a thin liquid film with which they exchange liquid and are thus very mobile. Here, using experiments, numerical simulations, and theory, I show that pendant drops sliding under a slightly tilted wet substrate can get stuck on topographic defects, despite their lack of contact line. Instead, this trapping has a gravito-capillary origin: liquid has to move up or down and the interface has to deform for the drop to pass the defect. I propose a semianalytical model for arbitrary substrate topographies that matches the trapping force observed, without any fitting parameter. I finally demonstrate how to harness this topography induced force to guide pendant drops on complex paths and expect it to be relevant for other contact line free systems.

The humble Petri dish is perhaps the simplest setting in which to examine the locomotion of swimming organisms, particularly those whose body size is tens of microns to millimeters. The fluid layer in such a container has a bottom no-slip surface and a stress-free upper boundary. It is of fundamental interest to understand the flow fields produced by the elementary and composite singularities of Stokes flow in this geometry. Building on the few particular cases that have previously been considered in the literature, we study here the image systems for the primary singularities of Stokes flow subject to such boundary conditions—the Stokeslet, rotlet, source, rotlet dipole, source dipole, and stresslet—paying particular attention to the far-field behavior. In several key situations, the depth-averaged fluid flow is accurately captured by the solution of an associated Brinkman equation whose screening length is proportional to the depth of the fluid layer. The case of hydrodynamic bound states formed by spinning microswimmers near a no-slip surface, discovered first using the alga Volvox, is reconsidered in the geometry of a Petri dish, where the power-law attractive interaction between microswimmers acquires unusual exponentially screened oscillations.

In thermochemical nonequilibrium processes, both the nonequilibrium between different kinds of internal energy of molecules and the non-Boltzmann (NB) energy state distribution significantly impact the dissociation rate coefficients. The conventional two-temperature (2-T) model fails to accurately portray these effects, especially the NB effect. Consequently, dissociation rate coefficients calculated by the 2-T model are inaccurate in simulating strong thermochemical nonequilibrium flow, resulting in a surface heat flux inconsistent with experimental data. This article investigates the influencing factor of the NB effect on the dissociation rate coefficient using the state-to-state (STS) method during the zero-dimensional heating process of ${\mathrm{N}}_2$ and ${\mathrm{O}}_2$. Based on this, we develop a fitting formula to precisely correct the NB effect. Furthermore, we propose an improved model by integrating this fitting formula with the single-group linear maximum entropy model, which considers only the effect of nonequilibrium between different kinds of internal energy. This improved model provides an accurate description of thermochemical nonequilibrium on the dissociation rate coefficients. To validate the effectiveness of the improved model, we simulate the nonequilibrium process following a normal shock. The results demonstrate that in strong thermochemical nonequilibrium flow, compared to the 2-T Park model, the maximum and average discrepancies between the translation temperatures calculated by the improved model and those by the STS method are reduced by more than 68% and 82%, respectively. Additionally, the results closely align with experimental data, indicating that the improved model can accurately depict the effect of thermal nonequilibrium on dissociation rate coefficients.

Distribution of the mean streamwise velocity in a turbulent boundary layer over a flat plate can be represented by the equation $U\sim {\eta }^{1/n}$, as was widely used in the past; $U$ and $\eta$ are the normalized velocity and the wall-normal distance, respectively. However, this $1/n\mathrm{th}$-power law is an empirical one. By incorporating either the Reynolds shear stress model of Wei et al. [J. Fluid Mech. 969, A3 (2023)], which is in terms of $U$ and the (normalized) wall-normal velocity ($V$), or a similar one in the boundary layer equations, it is found that $U$ and $V$ are related as $U^{(H+1)}\sim V^{(H-1)}$ in the outer region of a flat plate boundary layer; $H$ is the flow shape parameter. Along with the distribution of the wall-normal velocity ($V_w$) of Wei et al., the $1/n\mathrm{th}$-power law for $U$ is obtained by equating the derivative (with respect to $\eta$) of $V$ with that of $V_w$. Thus, this empirical power law seems to have a reasonable theoretical basis embedded in it.

Direct numerical simulations (DNSs) are among the most powerful tools for studying turbulent flows. Even though the achievable Reynolds numbers are lower than those obtained through experimental means, DNS offers a clear advantage: The entire velocity field is known, allowing for the evaluation of any desired quantity. This capability includes the computation of derivatives of all relevant terms. One such derivative provides the indicator function, which is the product of the wall distance and the wall-normal derivative of the mean streamwise velocity. This derivative may depend on mesh spacing and distribution, but it is extremely affected by the convergence of the simulation. The indicator function is crucial for understanding inner and outer interactions in wall-bounded flows and describing the overlap region between them. We find a clear dependence of this indicator function on the mesh distributions we examine, raising questions about classical mesh and convergence requirements for DNS and achievable accuracy. Within the framework of the logarithmic plus linear overlap region, coupled with a parametric study of channel flows and some pipe flows, sensitivities of extracted overlap parameters are examined. This study reveals a path to establishing their high-${\text{Re}}_{\tau }$ or nearly asymptotic values at modest Reynolds numbers, but larger than the ones used in this work, accessible by high-quality DNS with reasonable cost.

We investigate the dynamic evolution of heterogeneity in shear-thickening suspensions subjected to swirling excitation with a free surface. The uniform state of such a system may lose its stability when the oscillation frequency is above a threshold, and density waves spontaneously form [Shi et al., J. Fluid Mech. 984, A69 (2024)]. Here, we report a state where jammed clusters emerge in high-density regions of the density waves. The jammed cluster exhibits unique motion, creating downstream high-density regions distinct from the previously reported state of density waves. Additionally, theoretical calculations show that reducing suspension thickness lowers the frequency and global concentration $\mathrm{\Phi }$ threshold for the heterogeneity onset. Notably, the minimal $\mathrm{\Phi }$ for instability can be lower than the onset of discontinuous shear thickening transition. We also highlight the role of the free surface in cluster growth and persistence.

In the region close to the thermodynamic critical point and in the proximity of the pseudoboiling (Widom) line, strong property variations substantially alter the growth of modal instabilities, as revealed in Ren et al. [J. Fluid Mech. 871, 831 (2019)]. Here, we study nonmodal disturbances in the spatial framework using an eigenvector decomposition of the linearized Navier-Stokes equations under the assumption of locally parallel flow. To account for nonideality, a new energy norm is derived. Several heat transfer scenarios at supercritical pressure are investigated, which are of practical relevance in technical applications. The boundary layers with the fluid at supercritical pressure are heated or cooled by prescribing the wall and free-stream temperatures so that the temperature profile is either entirely subcritical (liquidlike), supercritical (gaslike), or transcritical (across the Widom line). The free-stream Mach number is set to ${10}^{-3}$. In the nontranscritical regimes, the resulting streamwise-independent streaks originate from the lift-up effect. Wall cooling enhances the energy amplification for both subcritical and supercritical regimes. When the temperature profile is increased beyond the Widom line, a strong suboptimal growth is observed over very short streamwise distances due to the Orr mechanism. Due to the additional presence of transcritical Mode II, the optimal energy growth at large distances is found to arise from an interplay between lift-up and Orr mechanism. As a result, optimal disturbances are streamwise-modulated streaks with strong thermal components and with a propagation angle inversely proportional to the local Reynolds number. The nonmodal growth is put in perspective with modal growth by means of an $N$-factor comparison. In the nontranscritical regimes, modal stability dominates regardless of a wall-temperature variation. In contrast, in the transcritical regime, nonmodal $N$ factors are found to resemble the imposition of an adverse pressure gradient in the ideal-gas regime. When cooling beyond the Widom line, optimal growth is greatly enhanced, yet strong inviscid instability prevails. When heating beyond the Widom line, optimal growth could be sufficiently large to favor transition, particularly with a high free-stream turbulence level.

We explore the potential for air bubble entrapment beneath micrometer-sized particles following immersion. This investigation employs theoretical, numerical, and experimental methodologies, with a focus on the wetting characteristics of both the particle and its substrate. These properties are crucial in determining the likelihood of entrapment and its impact on the particle's adhesion force to the substrate. The theoretical model provides the mathematical framework to account for the additional force exerted on the particle due to the entrapped bubble, while numerical calculations yield corresponding force values. The results underscore the significant influence of the wettability of both the particle and the substrate on this force. In support of findings of the numerical model, companion experiments were performed. The results demonstrate that the bubbles can indeed be entrapped at microscales underneath micrometric particles. Experimental measurements of detachment force reveal the substantial impact of these entrapped bubbles on the force required to detach particles from a surface. Specifically, the force appears notably higher when either the particle or the substrate, or both, exhibit hydrophobic characteristics. We highlight the alignment observed between numerical calculations and experimental results, while also examining and discussing any identified disparities and their root causes. Lastly, we propose an energy model that predicts the post-detachment configuration of the bubble, determining whether it remains attached to the particle, adheres to the substrate, or splits into daughter bubbles distributed across both surfaces. These findings hold significance for a wide range of industrial applications where the immersion of micrometer-sized entities, such as dirt or bacteria, is common during liquid-based cleaning processes.

Miscible multiphase flow in porous media is a key phenomenon in various industrial and natural processes, such as hydrogen storage and geological carbon sequestration. However, the parameters controlling the patterns of displacement and mixing in these flows are not completely resolved. This study delves into the effects of heterogeneity and inlet pressure on mixing and displacement patterns of low-viscosity miscible phase invasion into a high-viscosity resident phase, that is saturating a porous medium. The findings highlight the substantial influence of inlet pressures and heterogeneity levels in transitioning from uniform to fingering patterns at the pore scale. These phenomena are detectable at the Darcy scale, and their transition from a uniform front to finger formation is effectively marked through a modified Sherwood number. This modified Sherwood number links microscale patterns to physical properties such as velocity distribution, diffusion, and viscosity contrasts. Additionally, the study employs breakthrough curve (BTC) analysis to illustrate the role of higher heterogeneity and inlet pressure in broadening the fluid velocity distribution, leading to the fingering pattern. These research insights provide a nondimensional approach that scales the BTCs, and can serve future models of miscible phase flow in porous media, linking pore-scale dynamics with macroscale Darcy-scale observations.